IOSR Journal of Computer Engineering (IOSR-JCE)
e-ISSN: 2278-0661, p- ISSN: 2278-8727Volume 16, Issue 3, Ver. III (May-Jun. 2014), PP 01-07
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An Optimized Algorithm for Solving Combinatorial Problems
using Reference Graph
Raktim Chakraborty1,Saubhik Paladhi2,Sankhadeep Chatterjee3, Soumen
Banerjee4
1234
Hooghly Engineering & Technology CollegeVivekananda Road, Pipulpati, Hooghly-712103, West Bengal,
India
Abstract: In this paper a novel optimized graph referencing method is proposed to tackle combinatorial
problems with greater ease. The proposed algorithm is a modification of the existing backtracking algorithm. It
takes into account the advantage of backtrack algorithm while eliminating its disadvantages. Special care has
been taken to reduce the time complexity in avoiding frequent unsuccessful searches and in providing accurate
and fast solution to the targeted sub problem. The theoretical analysis and experimental results presented reveal
the superiority of proposed algorithm over the conventional algorithms.
Keywords:Backtracking, Combinatorial problem, Graph Referencing algorithm, Optimization, Sudoku.
I.
Introduction
A special branch of mathematics known as Combinatorics deals with the study of finite structure
including counting the structure of a given kind and size, depending upon criteria to be met and analysis and
constructs objectives meeting such criteria. Combinatorial problems find extensive application in fields of
mathematics, machine learning, AI, Auction Theory, Software Engineering etc. Backtracking is a general
algorithm for finding all or some solution to some combinatorial problems. However, Backtracking suffers
inherent disadvantages in terms of conjunction of a large processing time and memory space. It is here that the
proposed novel optimized graph referencing method established its superiority. The algorithm as proposed by
the authors, which is a modification of conventional Backtracking algorithm, not only solves efficiently
different combinatorial problems but does so in the least possible time.
Tackling various combinatorial problems is a subject of research worldwide. Different efficient
algorithms are being developed to this purpose. Xiao et. al. [1] designed a stepwise enumerative algorithm while
another algorithm based on GA was developed by Liu et. al. [2]. The Backtracking algorithm is reported to be
used to solve Sudoku, a combinatorial problem, by Xu [3]. The solution to the Sudoku problem was proposed
by Zhang [4] who intend to use the application development language AutoLISP embedded in AutoCAD for
this purpose. An altogether different algorithm to solve Sudoku was proposed by Zhao et. al. [5] while the same
was tackled with the help of cultural algorithm and GA optimization as reported by Mantere and Kolijonen [67]. The underlying science behind combinatorial problem was reported by Delahaye [8] while structural
engineering problems were solved using an innovative yet powerful search and optimization method by Lee and
Geem [9-10]. An improved and modified version of graph matching algorithm was developed by Cordella et. al.
[11]. The DLX algorithm (using Dancing Link) implements natural algorithm for exact cover problems in an
effective manner towards a faster solution as put forwarded by Knuth [12]. Several other combinatorial
problems was solved by a novel technique proposed by Hitotumatu and Noshita [13]. Thus the conventional
Backtracking algorithm finds extensive application in the field of science and technology. Thus the authors
inspired by the Backtracking algorithm took a challenge to design an absolute new algorithm eliminating the
disadvantages of the Backtracking algorithm for solving the various types of combinatorial problems, thereby
opening an absolute new window to the science and engineering fraternity for extensive research in the domain.
The present work will be very helpful in analysing and solving combinatorial problems and would be much
beneficial to the research community.
II.
Implementation of the Reference Graph
Several fields like Artificial Intelligence, Auction Theory, Software Engineering etc. have applications
related to Combinatorics. The target is to find the best or optimized result from a finite set of results. The
authors consider here a set of Sudoku problems as a typical example of combinatorial problems and aims to
solve it efficiently keeping in mind space and time complexity.
In generalized format a Sudoku has N number of row and columns such that the N x N grid is
partitioned into N number of √N x √N boxes or blocks. Each row, column and box has N number of cells with a
total number of cells in the grid being N². As per rule of Sudoku, a digit cannot exist in any cell of same row or
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An Optimized Algorithm for Solving Combinatorial Problems using Reference Graph
same column or same box more than once. Fig.1depicts a 4 x 4 Sudoku cell’s connection graph and it is
observed that there are total 16 nodes. If any two node Vi andVjare connected, there will be an edge between Vi
and Vj. Thus any element of the set can influence the other elements of the same. For instance in Fig..1 node 1 is
linked with the set A= {0,2,3,4,5,9,13} i.e. the digit in the cell 1 will not be placed in any node belongs to set A.
Moreover a cell under consideration is not connected with itself. Fig. 2 depicts a typicalReference Graph Matrix
for a Reference Graph depicted in Fig. 1.
The particular cell of the Sudoku matrix is generated by the relation:Index = (row * order of Sudoku grid + column)
Fig. 1. Reference Graph for 4 x 4
Fig. 2. Reference Graph Matrix
The main key factors for time management and efficiency of the solution are the Reference Graph &
the Referenced Graph matrix. The pre-produced Reference Graph matrix has been proposed in anefficient way
such that the problem solving time will be reduced as it places the appropriate digit in a cell by searching its
absence in all other cells which are connected with each other.
III.
The Proposed Algorithm
In this section an algorithm has been proposed that solves combinatorial problems. At first a Referenced Graph
matrix for this problem is to be produced. As the size of the matrix varies with the order of the problem
(problem with higher order causes higher order matrix production), the authors have decided to propose a
generalized algorithm which produces the connection matrix for a specific problem.
The generalized algorithm to produce Referenced Graph matrix is:-
* In algorithm order is the grid order of the problem. Graph [ ] [ ] is the matrix which is representing
the connection between the nodes and box no is the index of box under consideration. *
Algorithm: Graph Referenced Matrix ()
1)
repeat until k: = order² -1
do row ← k / order
col ← k mod order
count ← 0
2)
repeat for j: = 0 to order -1
do
p ← row * order + j
check if p = row*order + col then
continue
endif
Graph[k][count]←p
count←count+1
Next j
3)
repeat for i: = 0 to order -1
do
p←i*order+col
check if p = row * order + col then
continueend if
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An Optimized Algorithm for Solving Combinatorial Problems using Reference Graph
Graph[k][count]←p
count←count+1
Next i
4)
box no ← ((row / box order) * box order) + (col / box order)
5)
rowinitial ← (box no / √(order)) * √ (order)
6)
rowfinal ← rowinitial + (box order -1)
7)
columninitial ← (box no mod box order) * box order
8)
columnfinal ← columninitial + (box order -1)
9)
repeat for i: = rowinitial to rowfinal
10)
repeat for j: = columninitial to columnfinal
do p ← i * order + j
check if p = row * order + col or (i = row or j = col) then
continue
end if
do Graph [k][count]←p
count← count + 1
Next j
Next i
11)
k ← k+1
wend
The steps for solving are as follows:1)
Problem [x] [y] contains the total problem.
Problem [i] [j] = value (ith row and jth column cell)
Problem [i] [j] = 0 if ithrow andjth column cell is empty
2)
Empty [z] stores the indexes of empty cells. Index = (row * order of Sudoku grid + column) .
3)
Find the location of empty cell in Problem [ ] [ ] by following
s = Empty[m]
row = s / order
column = s % order where m ∈ [0,z-1]
So Problem [ row ] [ column ] is empty.
4)
Search from Graph [ s ] [ 0 ] to Graph [s] [count -1 ] to discard the elements which have already been
assigned .
5)
First valid digit is assigned to Problem [ row ] [ column ] .
If no valid digit is found then the above steps are repeated for m= m-1
6)
For successful placement of element the steps will be repeated for m=m+1 until all the empty cells are
filled with appropriate element to yield the solution.
7)
IV.
Implementation of Proposed Algorithm
To implement the algorithm the authors have decided to choose a 4 x 4 Sudoku problem with 5 clues in
Fig. 3 on which the operations will be done. Referenced Graph and Graph referenced matrix for a problem with
4 x 4 order have beenshown in Fig. 1 and Fig. 2. Graph referenced matrix can be derived from the above
algorithm .
After deriving the matrix the algorithm enter into the next stage where valid digits are assigned to cells.
The steps according to the algorithm are :
1)
Problem [ 4 ] [ 4 ] will store the content of the problem .
Problem [ 4 ] [ 4 ] =
{ 1, 0, 0, 4,
0, 4, 0, 0,
0, 0, 3, 1,
0, 0, 0, 0}
2)
Empty [ ] will store only the index of an empty cell . Relation for giving index in particular cell is
Index = (row * order of Sudoku grid + column) .
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An Optimized Algorithm for Solving Combinatorial Problems using Reference Graph
Here order of Sudoku grid is 4.
So Empty [ ] = {1,2,4,6,7,8,9,12,13,14,15} .
3)
Here Empty[ 0 ] = 1 .
So row = 1 / 4 = 0 and column = 1% 4 = 1 . From this we can say the 0 th row and 1st column element of the
problem is empty.
4)
Algorithm will search from Graph [ 1 ] [ 0 ] to Graph [ 1 ] [ 6 ] for the cell indexes which contain
digits. Assigned digits will be discarded for the cell under consideration. 1 st row of the matrix is
|0 2 3 5 9 13 4|
The list of row and column element
Row
Column
Value
( Problem[row]
[column] )
0/4 = 0
0%4=0
1
2/4=0
2%4=2
0
3/4=0
3 % 4 =3
4
5/4=1
5 % 4 =1
0
9/4=2
9 % 4 =1
0
13 / 4 = 3
13 % 4 =1
0
4/4=1
4 % 4 =0
0
5)
So 1 and 4 will be discarded from the list of valid number for Problem [ 0 ] [ 1 ] . So set of valid digit
for that cell is {2,3} .
6)
2 will be assigned to Problem [ 0 ] [ 1 ] i.e. Problem [ 0 ] [ 1 ] ← 2 which has been shown in Fig.4.
7)
Algorithm will now go for Empty [ 1 ] element and like the above steps it will try to find invalid digits
for Problem [Empty [ 1 ] / 4 ] [Empty [ 1 ] % 4 ] i.e. Problem [ 0 ] [ 2 ] and in this case it will be found that no
valid digit exist for that cell as all the digits are already in the cells which are connected with it.
8)
At this point backtrack procedure will take place by moving back to Problem [ 0 ] [ 1 ] and place next
valid digit for that cell .In this case Problem [ 0 ] [ 1 ] ← 3 i.e. 3 will be assigned to that cell (shown in Fig.5)
9)
Thus the algorithm will fill all the cells with valid digits. Solved problem has been shown in Fig.ure 6.
Fig. 3 :Main Problem
assigned
Fig. 4 : 2 is assigned
Fig. 5 : After backtracking 3 is
Fig. 6 : Complete solution
V.
Results and Discussion
The proposed Graph Referencing algorithm is tested upon randomly selected combinatorial problems
of various difficulty levels. The same set problems have also been solved using the conventional Backtracking
algorithm and the various comparative results are depicted in Fig. 7 to Fig. 12.
Fig. 7 depicts a comparative study on problem solution time of easy level combinatorial problems. A
plot of solution time vs. corresponding puzzles (Typical combinatorial problems under easy category) has been
depicted in the graph and the same has been re-arranged in ascending order of solution time. Analysis reveals
that much lesser solution time is required in the case of proposed Graph Referencing algorithm (Refer blue line
of the graph). Several corresponding points are marked with alphabets in the Fig.7 which indicates the
superiority of the proposed Graph Referencing algorithm over Backtracking algorithm. For example, for the 2 nd
combinatorial problem the point ‘B’ depicts a solution time of 0.001s using Graph Referencing algorithm
whereas solution time at point ‘A’ is 0.002s using Backtrack algorithm. The point ‘C’ for problem number ‘3’
depicts a solution time of 0.006s whereas point ‘D’ depicts a solution time of 0.004s, thereby clearly a gain in
solution time of approx. 0.002s using Graph Referencing algorithm is achieved. In the same way point ‘E’
depicts a solution time of 0.009s of 5th problem using Backtracking algorithm, which precedes the solution time
by 0.002s for solving the same by using Graph Referencing algorithm as depicted in point ‘F’.
Fig. 8 depicts similar analysis which has been done on a set of randomly chosen medium level
combinatorial problems. An analysis reveals that time requirement in solving a problem is pretty much lesser in
case of use of Graph Referencing algorithm than that for Backtracking algorithm. For instance point ‘A’ depicts
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An Optimized Algorithm for Solving Combinatorial Problems using Reference Graph
solution time 0.028s for 5th problem using backtracking algorithm while point ‘B’ depicts solution time of
0.022s for the same using graph Referencing algorithm. Hence the later one takes 0.006s lesser time in solving
the problem than the former one, clearly indicating the ingenuity in case of solving medium level combinatorial
problems.
Fig. 9 depicts the same analysis for a set of randomly chosen hard level combinatorial problems. The
point marked ‘B’, depicts solution time of 0.020s for the 4th problem of hard category using Graph Referencing
algorithm, which takes 0.004s less than the solution time of the same depicted in the graph as ‘A’ using
Backtracking algorithm. The analysis clearly indicates the superiority of the proposed Graph Referencing
algorithm over all three categories of combinatorial problems.
A set of 30 combinatorial problems comprising of various difficulty categories viz. easy, medium and
hard are chosen randomly and the solution time using proposed Graph Referencing algorithm of the 30
randomly chosen problems set is depicted in Fig. 10 using scattered points in blue colour. A trend line (Straight
line in blue) plotted from the scattered points is also depicted to indicate the overall nature of the time required
to solve these problems using Graph Referencing algorithm.
Fig. 11 depicts the solution time using scattered points in colour red of the same set of 30 randomly
selected combinatorial problems using Backtracking algorithm. The overall nature of the solution time has been
depicted using a red trend line using plotted scattered points depicting solution time of different combinatorial
problems. The equations of the trend lines are calculated to be
y = 0.001x – 0.0023
... (1)
y = 0.001x – 0.0052
... (2)
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An Optimized Algorithm for Solving Combinatorial Problems using Reference Graph
Further analysis of the graphs reveals that the straight lines are representing the overall nature of the
solution time using two different algorithms. A detail meaning of the two straight lines are given by equation (1)
& (2) what is actually depicted in Fig. 12.
A plot of the solution time for 30 different and randomly chosen problems of different standards ( viz.
easy, medium, & hard) is shown in Fig. 12 where the problems are solved using both Graph Referencing
algorithm and Backtracking algorithm. The trend lines (in red and blue dashed lines) formed out of the scattered
points are also depicted in the same graph. The equations of the trend lines are also shown in the graph. Thus it
is evident that both straight lines are parallel to each other owing to identical slope values m = 0.001. However
they differ in their intercept values c1 = -0.0023 and c2 = -0.0052. The graph clearly reveals that in all the cases
the trend line in red always remains below the trend line in blue which is a vivid indication of the fact that lesser
solution time is consumed for solving any type of combinatorial problems using Graph Referencing algorithm.
It thus establishes the superiority in performance of the proposed Graph Referencing algorithm in comparison to
the backtracking algorithm.
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An Optimized Algorithm for Solving Combinatorial Problems using Reference Graph
VI.
Conclusion
The present research work carried out by the authors in proposing the novel optimized Graph
Referencing algorithm is a modification over the common Backtracking algorithm in efficiently solving any
type of combinatorial problems using the least solution time possible. The results depicted in the paper will be
very useful for the researchers and mathematicians in handling and solving combinatorial problems with ease in
near future.
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